FOM: Review of Wolfram's Talk

[Steve Stevenson]

A Review of Wolfram's "The Foundations of Mathematics and Mathematica"

A videoconference talk given at the International Mathematica
Symposium in Hagenberg, Austria, 23 Aug 1999.

General Comment. This is a transcript of the talk. The talk itself is
very informal and therefore one has to be careful about reading too
much into the talk. I attempt to the best degree possible to give an
outline of the talk and do no commentary.

The talk is quite long and the printer from Netscape did not number
the pages. I do attempt to quote where ever possible to give
checkpoints. I hand-numbered the pages, so the page numbers should be
seen as approximations.

The Talk. Wolfram begins by alluding to some sort of project that
everyone is supposed to know about. He makes the comment, "I'm
basically trying to build a whole new science, certainly as big as any
of the existing sciences, like physics and chemistry and things, and
perhaps in some ways bigger." The basic point, he goes one, is "The
basic point of the science .... If you look at the history of science
for tha last 300 years... [the purpose]: to find mathematical
equations to represent things." His claim is that this hasn't worked
for biology and rhetorically asks, "...what else can one do?"

The matter: Why should those rules (of biology) be based on the
constructs that we happen to have in traditional mathematics?

His answer is "...rules embodied in computer programs, but can't
necessarily be easily represented in traditional mathematics." [p 2].

On page 2 he claims he has discussed these ideas with "Bruno
Buchberger, Dana Scott, Klaus Sutner and Greg Chaitin," who don't
always agree with him.

Page 3 starts a discussion of having "arbitrary computer programs" and
uses as an example of the genre, cellular automata. He then gives
several pages of cellular automata examples. His claim is that nature
is more bizarre than our mathematics since "...the rules we use in
engineering are essentially special ones that we've set up to be able
to do the tasks we want to do." He now says his new science leads to a
"major generalization of mathematics" by altering what kinds of rules
are available [p 8].

Again, rhetorically, "...if mathematics isn't dealing with geneuinely
arbitrary abstract rules, what exactly is it dealing with." He then
goes into a discussion of Babylonian versus Greek mathematics. [This
theme runs throughout. Feynman does a good job of this argument in The
Character of Natural Law, chapter 2.] His conclusion is "...the kinds
of rules that nature really seems to follow are ones that are pretty
easy to represent in simple computer programs, but almost impossible
to represent in traditional kind of arithmetic-and-geometry
mathematics" [p 9].

He then discusses how people choose the constructs they use in
mathematics. [Here comes the reduction to physics.] "I suppose it's
kind of weird: one usually thinks that mathematics is somehow more
general and more abstract than, say, physics.... .... But actually the
conclusion that I've come to is that that's not true, and that in fact
the rules in physics - while there are far fewer of them - are chosen
in a sense more arbitrarily - and are probably much more
representative of all possible rules than the ones that have typically
ended up studied in mathematics." On page 12 we get to the whole
thing: "In a sense I've been trying to reduce physics to mathemaitcs
to get a simple abstract structure that is, exactly, the whole
universe." He goes on to point out that the primitives of Mathematica
were chosen exactly correctly to develop this new science.

On page 15 we have his answer to what might be the limit of his new
idea for mathematics: transformation rules as they appear in
Mathematica. He goes on to talk about transformation rules and
eventually comes to semi-Thue systems. He goes on to explore some
ideas of transformation systems and criticizes the notation in
Principia Mathematica. He then goes on to criticize types in response
to a criticism of Mathematica; someone told him that Mathematica
couldn't succeed without types.

>From page 16-26, he gives examples of what he means and how
transformations support his ideas. He also shows examples of breadth
first evaluation of rules to illustrate the parallel nature of the
transformation process.  Eventually on page 27 he asks and answers the
question "....what's the analog of proof in this kind of system." His
answer: " .... A proof is like a path in this [parallel generated
network]. A proof is something that shows one that one particular
string can be derived from another string. Remember that the axioms
were the transformation rules here, ...." So what we need are
"super-transformation rules."

Page 29 has the comment: "Well, I have to admit something about the
models I've been using so far. They are a bit different from
whatpeople consider usually as mathematics, because they don't have
logic, they don't have a notion of logic explicitly built into
them. As I mentioned before, most of the traditional and proof-based
matheamtics is ultimately based on logic. I'm not sure that's the best
thing, but that's the way it is." He finishes the paper by using the
term "alternate mathematicses [sic]". Such are all these alternatives
outside "traditional [I guess he means 'what we use in classical
physics'] mathematics."